用扩散可积性研究复杂介质的微结构
Diffusion Spreadability as a Probe of the Microstructure of Complex Media Across Length Scales
摘要：
考虑溶质在两相之间的质量传递的时间依赖性问题，并假设溶质最初分布在一个相中（相2），而不存在于另一个相中（相1）。我们需要作为时间的函数的存在于相1中的总溶质的分数${\cal S}(t)$，我们称之为{ \it扩散性}，因为它是作为时间的函数的扩散信息的扩散性的量度。我们推导出了${\cal S}(t)$在任意欧氏空间维数$d$中的自协方差函数的精确的直接空间公式以及${\cal S}(t)$在谱密度方面的相应的傅立叶表示。我们推导出封闭形式的一般公式的短期和长期的行为的传播性的关键小规模和大规模的微观结构信息，分别。${\cal S}(t)$的长时间行为使人们能够区分从超均匀到非超均匀介质的整个微观结构谱。对于超均匀介质，无序与否，我们表明，“过剩”的铺展性，${\cal S}(\infty)-{\cal S}(t)$，衰减到其长时间的行为指数比任何非超均匀两相介质，“最慢”的反超均匀介质。隐形超均匀类的特点是具有最快的衰减速率在所有的几何不变的微观结构中的过度铺展性。此外，我们建立一个显着的连接之间的铺展性和离散几何中的一个突出问题，即，微观结构与“快速”铺展性也是那些可以从有效的“覆盖”的空间。我们还发现了迄今为止未被注意到的扩散性${\cal S}(t)$和NMR脉冲场梯度自旋回波幅度以及扩散MRI测量之间的显着联系。
英文摘要：
Consider the time-dependent problem of mass transfer of a solute between two phases and assume that the solute is initially distributed in one phase (phase 2) and absent from the other (phase 1). We desire the fraction of total solute present in phase 1 as a function of time, ${\cal S}(t)$, which we call the {\it spreadability}, since it is a measure of the spreadability of diffusion information as a function of time. We derive exact direct-space formulas for ${\cal S}(t)$ in any Euclidean space dimension $d$ in terms of the autocovariance function as well as corresponding Fourier representations of ${\cal S}(t)$ in terms of the spectral density. We derive closed-form general formulas for the short- and long-time behaviors of the spreadability in terms of crucial small- and large-scale microstructural information, respectively. The long-time behavior of ${\cal S}(t)$ enables one to distinguish the entire spectrum of microstructures that span from hyperuniform to nonhyperuniform media. For hyperuniform media, disordered or not, we show that the "excess" spreadability, ${\cal S}(\infty)-{\cal S}(t)$, decays to its long-time behavior exponentially faster than that of any nonhyperuniform two-phase medium, the "slowest" being antihyperuniform media. The stealthy hyperuniform class is characterized by an excess spreadability with the fastest decay rate among all translationally invariant microstructures. Moreover, we establish a remarkable connection between the spreadability and an outstanding problem in discrete geometry, namely, microstructures with "fast" spreadabilities are also those that can be derived from efficient "coverings" of space. We also identify heretofore unnoticed remarkable links between the spreadability ${\cal S}(t)$ and NMR pulsed field gradient spin-echo amplitude as well as diffusion MRI measurements.
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